Exploring q-Bernstein-Bézier surfaces in Minkowski space: Analysis, modeling, and applications

In this paper, we examine q-Bernstein-Bézier surfaces in Minkowski space-R13 with q as the shape parameter. These surfaces, a generalization of Bézier surfaces, have applications in mathematics, computer-aided geometric design, and computer graphics for the surface formation and modeling. We analyze the timelike and spacelike cases of q-Bernstein-Bézier surfaces using known boundary control points. The mean curvature and Gaussian curvature of these q-Bernstein-Bézier surfaces are computed by finding the respective fundamental coefficients. We also investigate the shape operator dependency for timelike and spacelike q-Bernstein-Bézier surfaces in Minkowski space-R13, and provide biquadratic and bicubic q-Bernstein-Bézier surfaces as illustrative examples for different values of the shape controlling parameter q.


Introduction
Mathematical models are used to describe a number of physical phenomena as well as the geometry of a structure.These models play an important role in understanding and designing a desired geometric structure: including architecture of 3-dimensional models of buildings, automotive and automobiles, aerospace technology to design an aircraft or a spacecraft, shipbuilding (geometry of floating vessels), geoscience for particular types of maps and the geometry of the molecular structures studied in Chemistry.The surfaces following certain constraint structure and the geometric properties of the surfaces find their applications in computer aided-manufacturing (CAM), computer aided-designs (CAD) and computer aided-geometric designs (CAGD).Curves and surfaces are the primary tools of (CAM/CAD/CAGD) systems, and they deliver information about the geometry and shape of the artifacts.For the construction of a curve or a surface, the appropriate form is its parametric representation.It has its dominance over the other representations of the curves or the surfaces when the prescribed boundary is given by the control points.The parametric representation is sufficiently flexible to control the shape of the curve and the surface and it is more convenient in its use to study the geometry of the surfaces rather than when a surface is expressed in non-parametric form.
A regular surface is a geometric object which restricts sharp edges and self-intersections and it is mathematically represented by a function of two parameters, usually called the surface parameters.In computer graphics (CG) and (CAD) systems, the curves and surfaces are usually expressed as the parametric-polynomials along with control points.The polynomial curves and polynomial surfaces depend on the bases functions.The restricted class of such curves and surfaces are the Be ´zier-curves and Be ´zier-surfaces [1].These curves and surfaces were initially used by Pierre-Be ´zier in designing the needed auto-surfaces.De-Casteljau algorithm is used for repeated linear interpolation of the control points for the desired Be ´zier curves and surfaces in Bernstein bases form.The Bernstein polynomials [2] as the weights of the Be ´zier curves and the surfaces control their shape for the prescribed network of the control-points.Be ´zier-surfaces have a set of algorithmic-properties which can be used to analyze and interpret the shapes.The Be ´zier-surfaces formed by the product of two Be ´zier-curves have the same properties as well.Be ´zier surfaces have found numerous applications [3,4] across various disciplines, particularly in optimization theory, where they are utilized to find minimal Be ´zier surfaces, serving as extremals constrained by energy functionals in both Euclidean-R 3 and Minkowski space-R 3  1 [5][6][7][8][9][10][11][12].These surfaces show promise as candidates for optimization studies.This variational instance can be observed when deriving the EFEs as the result of vanishing variation of the Einstein-Hilbert action [13][14][15].These surfaces can be further analyzed for the vanishing mean curvature condition that leads to the PDEs, which can be utilized to uncover the symmetries of the surfaces.For instance, discussions related to Lie symmetries can be found in the works [16][17][18][19][20].
Based on Bernstein-polynomials, several generalized-versions of the Be ´zier-surfaces are in use.One of such generalization of the classical-Bernstein polynomial, q-Bernstein polynomial (q, an integer) is admitted by Phillips [21,22] (for its basic properties) and Oruc and Phillips [22] for the parametric representation of the curves.Other representation of the q-Bernsteinpolynomials is by Kim [23], given as a linear-combination of higher-order polynomials.Kim representation of the q-Bernstein-polynomials is referred to as q-extension of the Bernsteinpolynomials and they differ from Phillips representation of q-Bernstein-polynomials.The Kim [23] version used for q-Bernstein-polynomials facilitates one to find the derivatives in terms of lower-degree polynomials.Simsek and Acikgoz [24] addressed a new approach for generating new functions which produces the q-Bernstein type-polynomials.This construction differs from many of the previous-constructions in that they all used a recursion-formula.Sometimes the constructed Be ´zier-curves and Be ´zier-surfaces need to have their shapes changed in order to suit the requirements of our model.Apart from their use in optimization theory, the shape operator properties of these surfaces are also important when we introduce basis-functions with shape-control parameters.Khan [25] introduced a new class of curves and surfaces recognized as (p, q)-Bernstein-Be ´zier curves and (p, q)-Bernstein-Be ´zier surfaces [26] and surfaces that are an extension of q-Bernstein-Be ´zier curves and q-Bernstein-Be ´zier surfaces respectively.With the help of the parameters p and q, curves and surfaces can be modified in shape without changing the position of control-points.They also discussed some of its properties like partition of unity, end point property and non-negativity.Ahmad et al. [27] have discussed a computational scheme as a model for a quasi-minimal surface for the q-Bernstein Be ´zier-surface in the R 3 -Euclidian-space which can be extended for the Minkowski space or alternatively the equivalent Euler Lagrange equation, the partial differential equation can be solved using the technique [28][29][30].
On the other hand, H. Minkowski [31] made an initial contribution to address the geometry of the objects moving in four dimensional spacetime in relativity (special and general relativity), in which space coordinates and time coordinates are mixed together and are not separable in the Riemannian and pseudo-Riemannian metrics.The Minkowski space comprises three space coordinates (namely x, y, z) and the time coordinate (ct-coordinate) is taken as the fourth coordinate.However, there is resemblance between the Euclidean and the Minkowski space while defining the distance concept.This enables one to find the surfaces in 3-dimensional Minkowski space.The metric element for the three-dimensional Minkowski space [32] is , where (x 1 , x 2 , x 3 ) are the canonical-coordinates in Minkowski space-R 3  1 .The Lorentz-Minkowski metric in Minkowski space-R 3 1 separates the regions into three types of vectors, they are timelike-vectors, lightlike-vectors and spacelike-vectors.In the light-like region of the Minkowski space, the null-vectors, pseudo-null-curves, null-curves, marginally trapped surfaces, B-scrolls pose, measuring the angular displacement is obscure.Many others have studied and analyzed timelike and spacelike surfaces in Minkowski space-R 3  1 in different disciplines of interest in science.Treibergs [33] has investigated spacelike hypersurfaces of constant mean-curvature in Minkowski space-R 3  1 .For timelike-surfaces with a defined Gauss-map, Aledo et al. [34] have examined Lelievvre-type representation.In Minkowski space-R 3  1 , Abdel-Baky and Abd-Ellah [35] investigated both (spacelike or timelike) governed W-surfaces.Brander et al. [36] used the non-compact real form SU to construct spacelike constant-mean curvature surfaces in Minkowski space-R 3 1 .Lin [37] studied the impacts of curvature restrictions on the timelike-surfaces in Minkowski space-R 3  1 that are convex in the same way as are the surfaces in the Euclidean space-R 3 .Kossowski [38] obtained zero-mean curvature surface constraints in Minkowski space-R 3  1 .Georgiev [39] found sufficient conditions for the spacelike Be ´zier surfaces.Kuşak Samancı and Celik [40] analyzed the geometric characteristics such as shape operator, Gauss curvature and mean curvature of the Be ´zier surfaces in Minkowski space.In addition, related studies by Ceylan [41] focus on the geometry of Be ´zier curves in Minkowski space.Kılıc ¸oglu and Şenyurt [42] investigate methods for determining Be ´zier curves when their derivatives are given.Kılıc ¸oğlu and Yurttanc ¸ıkmaz [43] explore Be ´zier curve representation of exponential curves.These studies provide valuable insights into various aspects of differential geometry, including our investigation into q-Bernstein surfaces.
In this work, we investigate a specialized and important class of surfaces that are utilized in computer graphics, the q-Bernstein-Be ´zier surface in Minkowski space-R 3  1 .The objective is to determine the fundamental coefficients for the Gaussian curvature, mean curvature, and shape operator of timelike and spacelike q-Bernstein-Be ´zier surfaces in the Minkowski space.The obtained results are then applied to the corresponding shape operator of the biquadratic and bicubic (timelike/spacelike) q-Bernstein-Be ´zier surfaces in the Minkowski space, using q as the shape controlling parameter, to demonstrate the scheme.
The paper is organized as follows: Section 2 gives some preliminaries related to our work to make this paper self contained.In section 3, we demonstrate the shape operators of the nondegenerate cases of q-Bernstein-Be ´zier surfaces (qbbs) in Minkowski space-R 3  1 .For the illustration of the scheme work of section 3, we provide numeric work related to the construction of biquadratic and bicubic (timelike/spacelike) q-Bernstein-Be ´zier surfaces (qbbs) in the section 4. Section 5 comprises final remarks and a glimpse of future work.

Preliminaries
In this section, we review basic concepts that will be later used in the work.In the Minkowski space, R 3 1 ¼ ðR 3 ; φ L ð; ÞÞ, the Lorentzian-inner product of the two vectors α and β with the metric signature (2,1), is defined as, where α = (α 1 , α 2 , α 3 ) and β = (β 1 , β 2 , β 3 ) are the vectors in 3-dimensional space.A 3-vector β in Minkowski space-R The timelike and the spacelike vectors are the non-degenerate vectors in the Minkowski space-R 3 1 .The cross-product of two vectors α and β in Minkowski space-R 3 1 is where ^L denotes the Lorentzian cross product in the Minkowski space-R 3 1 .Let M be a surface represented by a regular parameterized surface s : Let T P ðMÞ be the tangent plane at a point P on the surface M spanned by the tangent vectors to the coordinate curves s(u, v 0 ) and s(u 0 , v).Then the unit normal N(u, v) at the point Pðsðu; vÞÞ on the surface M is the vector field, given by The first fundamental form on the plane T P ðMÞ at the point P of the surface M corresponds to the matrix, ; where det ðoÞ ¼ EG À F 2 ; ð2:5Þ and E, F, G are the coefficients of the first-fundamental form of the surface s(u, v) defined by for the non-degenerate surfaces (timelike or the spacelike surface) in Minkowski space-R 3 1 .For a spcelike surface, det (ω) > 0 and for a timelike surface, det (ω) < 0. Non-degenerate surfaces (a timelike or a spacelike surface) in Minkowski space-R 3  1 are characterized by the term φ L ðN; NÞ ¼ Z.For a spacelike-surface, the normal N is a timelike vector as the tangent-plane T P ðMÞ is spacelike, and thus, φ L ðN; NÞ ¼ Z ¼ À 1, whereas for a timelike-surface, the normal N is a spacelike-vector as the tangent-plane T P ðMÞ is timelike, and thus in this case, φ L ðN; NÞ ¼ Z ¼ 1.So that, Lorentzian cross-product (Eq (2.3)) of vectors s u and s v yields, ks u ^L s v k ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi À Z det o p : ð2:7Þ The coefficients e, f, g of second-fundamental form of the surface M on a tangent-plane T P ðMÞ can be computed by using the Lorentzian-inner product (Eq (2.1)) of the unit normal vector N with that of the second order partial derivatives s uu , s uv , s vv of the regular parameterized surface ð2:9Þ corresponds to the shape operator of the surface s(u, v).From the above matrix (2.9) related to the shape operator of the surface, one can compute the mean-curvature H and the Gauss-curvature K of the surfaces in its non-degenerate case (a spacelike or a timelike surface) as follows, where Q n;q | ðvÞ is the n th degree q-Bernstein-polynomial, for the shape controlling parameter q, . In particular, commonly known quadratic and cubic q-Bernstein-Be ´zier curves (qbbc) determined from the Eq (2.14) (for n = 2 and n = 3 respectively) are shown in Fig 1 .Note that, q-Bernstein-polynomials of degree n = 5 can be computed from Eq (2.15), shown in Fig 2 for q, for q = 0.2, 0.4, 0.6, 0.8, 1.
Let [ | be the respective control-points of the curve obtained from Eq (2.14) for the q-Bernstein-Be ´zier curve in the Euclidian space-R 3 .Then, its first order derivative w.r.t. the curve parameter v comes out as, where ðQ m;q | ðvÞÞ v represents the derivative of q-Bernstein-polynomial Q m;q | ðvÞ with respect to v and it is defined by (for the detail see the ref.[27], Eqs (3.10) to (3.27)) It is to be remarked that the derivative of the q-Bernstein-polynomial appears as the Fifth degree q-Bernstein polynomials for the shape controlling parameter q, for q = 0.2, 0.4, 0.6, 0.8, 1.
https://doi.org/10.1371/journal.pone.0299892.g002polynomials of lesser degree (see Eq (2.18)) and the control points as the forward differences in the respective coordinates (see Eq (2.19)).Definition 2.4.A q-Bernstein-Be ´zier surface s(u, v) is the tensor-product of q-Bernstein bases-functions Q m;q | ðuÞ and Q n;q ' ðvÞ (Eq (2.15)) along with the control-points [ 00 , [ 01 ,. .., [ mn in Euclidian space-R 3 and it is represented as The coordinate curves (usually called u-parameter or the v-parameter curves) on the q-Bernstein-Be ´zier surface s(u, v) can be determined by choosing one of the surface parameters as the constant.They are in the form s(u, v 0 ) or s(u 0 , v).The coordinate curves s(u, 0), s(u, 1), s(0, v) and s(1, v) are the q-Bernstein-Be ´zier curves (compare it with Eq (2.14)).The coordinate curves s(u, 0), s(u, 1), s(0, v) and s(1, v) comprise the four edges of the q-Bernstein-Be ´zier surfaces (qbbs) along with the endpoint interpolation at the corner-points, It is to be noted that a q-Bernstein-Be ´zier surface (qbbs) is invariant under a three dimensional affine transformation L by virtue of the following equality Now, we present several results pertaining to the partial derivatives of the q-Bernstein-Be ´zier surface s(u, v).First order partial-derivative s u (u, v) of the q-Bernstein-Be ´zier surface s(u, v) (Eq (2.20)), with respect to the surface parameter u is where In the similar manner, the partial derivative of the first order of qbbs, q-Bernstein-Be ´zier surface, w.r.t the surface parameter v is where One can compute the partial derivatives of first order of the q-Bernstein-Be ´zier surface s(u, v) with surface parameter u and v, at the minimum-point (u, v) = (0, 0), from the Eqs (2.23)-(2.26), From Eq (2.23), partial derivative of second order of q-Bernstein-Be ´zier surface s(u, v) w.rt.u is where Similarly, using Eq (2.23), mixed partial derivative of second order, of q-Bernstein-Be ´zier surface s(u, v) is where, Now, from the Eq (2.25), partial derivative of second order, of q-Bernstein-Be ´zier surface (qbbs) w.r.t.parameter v is where We can find now s uu (0, 0), s uv (0, 0) and s vv (0, 0), second-order partial derivatives of the q-Bernstein-Be ´zier surface s(u, v) with respect to the surface parameters u and v at the minimum point (u, v) = (0, 0).From Eqs (2.29) and (2.30), we find that whereas s uv (0, 0) can be obtained from the Eqs (2.31) and (2.32), 1Þ 00 ; ð2:36Þ and s vv (0, 0), from the Eqs (2.33) and (2.34) is The aforementioned derivatives are computed for the upcoming section 3 with the aim of determining the shape operator in non-degenerate cases (timelike/spacelike) of q-Bernstein-Be ´zier surfaces (qbbs) in Minkowski space.These derivatives can then be used in the framework presented in section 4, which focuses on numerical computations for the biquadratic and bicubic (timelike/spacelike) q-Bernstein-Be ´zier surfaces (qbbs).

Results and discussion
In this section, we find the metric coefficients of q-Bernstein-Be ´zier surface (qbbs) for the two cases, timelike and spacelike surfaces, by generalizing q-Bernstein-Be ´zier surface (qbbs) in Minkowski space-R 3 1 .This enables us to find the Gauss-curvature and mean-curvature of the q-Bernstein-Be ´zier surface (qbbs) and the corresponding matrix form of the shape operator in Minkowski space-R 3  1 .Definition 3.1.A q-Bernstein-Be ´zier surface (qbbs), s(u, v) (where |;'¼0 , as the tensor product of q-Bernstein-bases functions Q m;q | ðuÞ and Q n;q ' ðvÞ is indicated in the Eq (2.20).We are interested in the non-degenerate cases of the q-Bernstein-Be ´zier surfaces (qbbs).In the Minkowski space-R 3  1 , q-Bernstein-Be ´zier surface, s(u, v) given by Eq (2.20) is said to be timelike if φ L ðN; NÞ ¼ 1 and spacelike if φ L ðN; NÞ ¼ À 1, where N is the unit normal to the surface.
Theorem 3.1.It can readily be seen that coefficients E, F and G of the first fundamental form of the (timelike/spacelike) q-Bernstein-Bézier surface s(u, v) in Minkowski space-R 3  1 can be computed from the Eq (2.6).Thus the coefficient E (from the Eq (2.6)) of the first-fundamental form of the q-Bernstein-Bézier surface (qbbs), along with the first-order partial derivative s u (u, v) of the q-Bernstein-Bézier surface (qbbs) from the Eq (2.23) and the Lorentzian-inner product defined in Eq (2.1), is given by E ¼ φ L ðs u ðu; vÞ; s u ðu; vÞÞ In the similar way other coefficients F and G can be computed and they are, In the Minkowski space-R 3 1 , the coefficients E, F, G of the first-fundamental form of the (timelike/spacelike) q-Bernstein-Bézier surface (qbbs) can be obtained from the Eqs (3.1)-(3.3),at the minimum point (u, v) = (0, 0) The components (μ 1 , μ 2 , μ 3 ) of the numerator μ ¼ s u ^L s v of the unit normal N to the q-Bernstein-Be ´zier surface (qbbs) are, ð3:9Þ It is to be noted that in the results below, η = 1 for the timelike q-Bernstein-Be ´zier surface s(u, v) and η = −1 for the spacelike q-Bernstein-Be ´zier surface s(u, v).
Theorem 3.2.In the Minkowski space-R 3 1 , the normal vector-field N to the non-degenerate (timelike/spacelike) q-Bernstein-Bézier surface s(u, v) is given by, N ¼ μ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi À Z φ L ðμ; μÞ p ; ð3:10Þ where μ ¼ s u ^L s v .The components (μ 1 , μ 2 , μ 3 ) of μ are given in Eqs (3.7)-(3.9),and η = 1, for the timelike q-Bernstein-Bézier surface (qbbs) and η = −1 if it is a spacelike-surface.Proof.In the Minkowski space-R 3 1 , the unit normal N to the tangent plane T P ðMÞ at the point P on the surface s(u, v) spanned by the tangent vectors s u and s v to the coordinate curves on the (timelike/spacelike) q-Bernstein-Be ´zier surfaces s(u, v), (by Lorentzian-cross product (as defined by the Eq (2.3) of the tangent vectors s u and s v ), is Plugging the values of the s u and s v from the Eqs (2.23) and (2.25) in the Eq (3.11), we get where φ L ðN; NÞ ¼ Z for both the timelike and spacelike surfaces.For timelike q-Bernstein-Be ´zier surface s(u, v), φ L ðN; NÞ ¼ 1 whereas for the spacelike, φ L ðN; NÞ ¼ À 1 and the norm ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi jE G À F 2 j p ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi À Z det ðoÞ p : ð3:13Þ Thus, the surface-normal Eq (3.12), by virtue of the Eqs (2.1) and (3.13) can be written as N ¼ ðm 1 ; m 2 ; m 3 Þ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ; m 2 ; m 3 Þ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Proof.Note that the determinant det (ω) = EG − F 2 of the first-fundamental form of the (timelike/spacelike) q-Bernstein-Be ´zier surface (qbbs) in Minkowski space-R 3 1 is given by the Eq (2.5).Substituting the fundamental coefficients E, F, G (given by Eqs (3.1)-(3.3)) in this Eq (2.5), we find that The terms in above Eq (3.17) when compared with that of the components (μ 1 , μ 2 , μ 3 ) (Eqs (3.7) to (3.9)) of the vector μ reduces it to the result stated in Eq (3.16).Corollary 3.3.1.The determinant det (ω) of the corresponding matrix ω of the first-fundamental form can be obtained directly from the fundamental coefficients (3.4) and it follows that the determinant det (ω) in Eq (3.18) can be rewritten in the form as stated in Eq (3.16).
Theorem 3.4.The coefficients (2.8) of the second fundamental form of the (timelike/spacelike) q-Bernstein-Bézier surface (qbbs) can be written in following form e ¼ ð ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Q mÀ 2;q | ðuÞQ n;q ' ðvÞ[ ð2;0Þ |' ; further simplification of Eq (3.24) results in, Q mÀ 2;q | ðuÞQ n;q ' ðvÞx ð2;0Þ |' m 3 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi and thus the Eq (3.25) can be written in the following simpler useful form Q mÀ 2;q | ðuÞQ n;q ' ðvÞx ð2;0Þ |' m 3 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Similarly, the coefficients f and g (of Eq (2.8)) of the second-fundamental form can be found by using the second-order partial derivative s uv (u, v) and s vv (u, v) of Eqs (2.31) and (2.33) respectively, and the normal vector N, Eq (3.12).where, μ = (μ 1 , μ 2 , μ 3 ) and μ 1 , μ 2 and μ 3 are given by the Eq (3.19).

Theorem 3.5. We can find the Gaussian
of the (timelike/spacelike) q-Bernstein-Bézier surface s(u, v) in Minkowski space by using the fundamental coefficients given in the Theorem 3.1 and the Theorem 3.4.It follows that the Gaussian-curvature K of the q-Bernstein-Bézier surface (qbbs) can be written as, We may skip the proof since the computations involved are straightforward.Corollary 3.5.1.Thus, the Gaussian-curvature (from the above Eq (3.28)) and the mean-curvature (from the Eq (3.29)) of the (timelike/spacelike) q-Bernstein-Bézier surface s(u, v) in the Minkowski space-R 3  1 , at the minimum-point (u, v) = (0, 0) come out to be, ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi À Zφ " # corresponding to the shapeoperator of the (timelike/spacelike) q-Bernstein-Bézier surface s(u, v) in Minkowski space-R 3 1 are, b 11 ¼ À 1 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi and, b 22 ¼ 1 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi Proof.The matrix corresponding to the shape-operator of the (timelike/spacelike) q-Bernstein- Note that the fundamental coefficients are given in the statements of the Theorem 3.1 and Theorem 3.4.Thus, the matrix element b 11 ¼ e G À f F E G À F 2 of the above matrix V is b 11 ¼ À 1 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Theorem 3.7.We can adopt the alternative approach for convenience, in order to find the Gaussian-curvature and the mean-curvature of the (timelike/spacelike) q-Bernstein-Bézier surface s(u, v) by using the shape-operator of the surface in Minkowski space-R 3  1 , they are ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Proof.As mentioned above in the statement that the Gaussian-curvature and mean-curvature of the (timelike/spacelike) q-Bernstein-Be ´zier surface s(u, v) in Minkowski space-R 3 1 can be found by using the matrix-coefficients corresponding to the shape-operator.The Gaussiancurvature (K = η det (V) = η(b 11 b 22 − b 12 b 21 )) of the (timelike/spacelike) q-Bernstein-Be ´zier surface (qbbs), by virtue of the shape operator matrix coefficients (3.32)-(3.35)),turns up Plugging the Eq (3.20) in above Eq (3.40) for the Gaussian curvature K in the following form ; ð3:41Þ a little simplification in above equation, reduces it to Similarly, the mean curvature, Þ, can be obtained by utilizing the shape operator matrix coefficients (3.32)-(3.35)and it takes the form, ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi À Zφ 3 L ðμ; μÞ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ð3:43Þ Corollary 3.7.1.The matrix-coefficient b 11 (Eq (3.32)) of the shape operator matrix V of the (timelike/spacelike) q-Bernstein-Bézier surface s(u, v) in the Minkowski space-R 3  1 , at the point (u, v) = (0, 0) turns up, b 11 ¼ À 1 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi À Z φ Theorem 3.8.The shape-operator matrix-coefficients of the matrix V of the (timelike/spacelike) q-Bernstein-Bézier surface s(u, v) can be used to find the Gaussian-curvature in Minkowski space-R 3  1 at the min-point (u, v) = (0, 0).It follows that Similarly, we can find the mean-curvature of q-Bernstein-Bézier surface s(u, v) in Minkowski space-R 3 1 at the min-point (u, v) = (0, 0) and it is, ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi À Zφ ð3:49Þ Proof.The Gaussian-curvature and the mean-curvature at the minimum-point (u, v) = (0, 0) of the timelike and the spacelike-surface q-Bernstein-Be ´zier surface s(u, v) are calculated by substituting the coefficient values in the Corollary 3.7.1 into the Eqs (3.38) and (3.39)). 4 The numeric examples of (Timelike/Spacelike) q-Bernstein-Be ´zier surfaces In this section, the shape operator dependence of timelike and spacelike q-Bernstein-Be ´zier surfaces (qbbs) in the Minkowski space-R 3 1 discussed in the above section 3 is implemented for the biquadratic and bicubic (timelike/spacelike) q-Bernstein-Be ´zier surfaces (qbbs).These surfaces serve as the illustrative examples as the timelike and spacelike surfaces for different values of the shape controlling parameter q.

Conclusion
In this paper, we present a family of Be ´zier surfaces called q-Bernstein-Be ´zier surfaces, in R 3 1 -Minkowski space.We investigate the shape operators for the non-degenerate cases of these surfaces and provide illustrative examples of biquadratic and bicubic degenerate q-Bernstein-Be ´zier surfaces.The techniques used to find the shape operators of q-Bernstein-Be ´zier surfaces in Minkowski space are promising for further analysis in the field of differential geometry.The findings of this study can be useful in optimizing the shape of these surfaces to fit the requirements of a computational model for a surface, particularly in areas such as computer-aided geometric design and computer graphics.